Properties

Label 245.2.a.d.1.2
Level $245$
Weight $2$
Character 245.1
Self dual yes
Analytic conductor $1.956$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 245 = 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 245.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(1.95633484952\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \(x^{2} - x - 4\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 35)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.56155\) of defining polynomial
Character \(\chi\) \(=\) 245.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.56155 q^{2} +2.56155 q^{3} +0.438447 q^{4} -1.00000 q^{5} +4.00000 q^{6} -2.43845 q^{8} +3.56155 q^{9} +O(q^{10})\) \(q+1.56155 q^{2} +2.56155 q^{3} +0.438447 q^{4} -1.00000 q^{5} +4.00000 q^{6} -2.43845 q^{8} +3.56155 q^{9} -1.56155 q^{10} +2.56155 q^{11} +1.12311 q^{12} -4.56155 q^{13} -2.56155 q^{15} -4.68466 q^{16} +4.56155 q^{17} +5.56155 q^{18} -1.12311 q^{19} -0.438447 q^{20} +4.00000 q^{22} -5.12311 q^{23} -6.24621 q^{24} +1.00000 q^{25} -7.12311 q^{26} +1.43845 q^{27} -5.68466 q^{29} -4.00000 q^{30} -2.43845 q^{32} +6.56155 q^{33} +7.12311 q^{34} +1.56155 q^{36} +6.00000 q^{37} -1.75379 q^{38} -11.6847 q^{39} +2.43845 q^{40} +3.12311 q^{41} +9.12311 q^{43} +1.12311 q^{44} -3.56155 q^{45} -8.00000 q^{46} -3.68466 q^{47} -12.0000 q^{48} +1.56155 q^{50} +11.6847 q^{51} -2.00000 q^{52} +3.12311 q^{53} +2.24621 q^{54} -2.56155 q^{55} -2.87689 q^{57} -8.87689 q^{58} +4.00000 q^{59} -1.12311 q^{60} +9.36932 q^{61} +5.56155 q^{64} +4.56155 q^{65} +10.2462 q^{66} -6.24621 q^{67} +2.00000 q^{68} -13.1231 q^{69} +8.00000 q^{71} -8.68466 q^{72} -4.24621 q^{73} +9.36932 q^{74} +2.56155 q^{75} -0.492423 q^{76} -18.2462 q^{78} -6.56155 q^{79} +4.68466 q^{80} -7.00000 q^{81} +4.87689 q^{82} -4.00000 q^{83} -4.56155 q^{85} +14.2462 q^{86} -14.5616 q^{87} -6.24621 q^{88} -7.12311 q^{89} -5.56155 q^{90} -2.24621 q^{92} -5.75379 q^{94} +1.12311 q^{95} -6.24621 q^{96} +14.8078 q^{97} +9.12311 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{2} + q^{3} + 5q^{4} - 2q^{5} + 8q^{6} - 9q^{8} + 3q^{9} + O(q^{10}) \) \( 2q - q^{2} + q^{3} + 5q^{4} - 2q^{5} + 8q^{6} - 9q^{8} + 3q^{9} + q^{10} + q^{11} - 6q^{12} - 5q^{13} - q^{15} + 3q^{16} + 5q^{17} + 7q^{18} + 6q^{19} - 5q^{20} + 8q^{22} - 2q^{23} + 4q^{24} + 2q^{25} - 6q^{26} + 7q^{27} + q^{29} - 8q^{30} - 9q^{32} + 9q^{33} + 6q^{34} - q^{36} + 12q^{37} - 20q^{38} - 11q^{39} + 9q^{40} - 2q^{41} + 10q^{43} - 6q^{44} - 3q^{45} - 16q^{46} + 5q^{47} - 24q^{48} - q^{50} + 11q^{51} - 4q^{52} - 2q^{53} - 12q^{54} - q^{55} - 14q^{57} - 26q^{58} + 8q^{59} + 6q^{60} - 6q^{61} + 7q^{64} + 5q^{65} + 4q^{66} + 4q^{67} + 4q^{68} - 18q^{69} + 16q^{71} - 5q^{72} + 8q^{73} - 6q^{74} + q^{75} + 32q^{76} - 20q^{78} - 9q^{79} - 3q^{80} - 14q^{81} + 18q^{82} - 8q^{83} - 5q^{85} + 12q^{86} - 25q^{87} + 4q^{88} - 6q^{89} - 7q^{90} + 12q^{92} - 28q^{94} - 6q^{95} + 4q^{96} + 9q^{97} + 10q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.56155 1.10418 0.552092 0.833783i \(-0.313830\pi\)
0.552092 + 0.833783i \(0.313830\pi\)
\(3\) 2.56155 1.47891 0.739457 0.673204i \(-0.235083\pi\)
0.739457 + 0.673204i \(0.235083\pi\)
\(4\) 0.438447 0.219224
\(5\) −1.00000 −0.447214
\(6\) 4.00000 1.63299
\(7\) 0 0
\(8\) −2.43845 −0.862121
\(9\) 3.56155 1.18718
\(10\) −1.56155 −0.493806
\(11\) 2.56155 0.772337 0.386169 0.922428i \(-0.373798\pi\)
0.386169 + 0.922428i \(0.373798\pi\)
\(12\) 1.12311 0.324213
\(13\) −4.56155 −1.26515 −0.632574 0.774500i \(-0.718001\pi\)
−0.632574 + 0.774500i \(0.718001\pi\)
\(14\) 0 0
\(15\) −2.56155 −0.661390
\(16\) −4.68466 −1.17116
\(17\) 4.56155 1.10634 0.553170 0.833069i \(-0.313418\pi\)
0.553170 + 0.833069i \(0.313418\pi\)
\(18\) 5.56155 1.31087
\(19\) −1.12311 −0.257658 −0.128829 0.991667i \(-0.541122\pi\)
−0.128829 + 0.991667i \(0.541122\pi\)
\(20\) −0.438447 −0.0980398
\(21\) 0 0
\(22\) 4.00000 0.852803
\(23\) −5.12311 −1.06824 −0.534121 0.845408i \(-0.679357\pi\)
−0.534121 + 0.845408i \(0.679357\pi\)
\(24\) −6.24621 −1.27500
\(25\) 1.00000 0.200000
\(26\) −7.12311 −1.39696
\(27\) 1.43845 0.276829
\(28\) 0 0
\(29\) −5.68466 −1.05561 −0.527807 0.849364i \(-0.676986\pi\)
−0.527807 + 0.849364i \(0.676986\pi\)
\(30\) −4.00000 −0.730297
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) −2.43845 −0.431061
\(33\) 6.56155 1.14222
\(34\) 7.12311 1.22160
\(35\) 0 0
\(36\) 1.56155 0.260259
\(37\) 6.00000 0.986394 0.493197 0.869918i \(-0.335828\pi\)
0.493197 + 0.869918i \(0.335828\pi\)
\(38\) −1.75379 −0.284502
\(39\) −11.6847 −1.87104
\(40\) 2.43845 0.385552
\(41\) 3.12311 0.487747 0.243874 0.969807i \(-0.421582\pi\)
0.243874 + 0.969807i \(0.421582\pi\)
\(42\) 0 0
\(43\) 9.12311 1.39126 0.695630 0.718400i \(-0.255125\pi\)
0.695630 + 0.718400i \(0.255125\pi\)
\(44\) 1.12311 0.169315
\(45\) −3.56155 −0.530925
\(46\) −8.00000 −1.17954
\(47\) −3.68466 −0.537463 −0.268731 0.963215i \(-0.586604\pi\)
−0.268731 + 0.963215i \(0.586604\pi\)
\(48\) −12.0000 −1.73205
\(49\) 0 0
\(50\) 1.56155 0.220837
\(51\) 11.6847 1.63618
\(52\) −2.00000 −0.277350
\(53\) 3.12311 0.428992 0.214496 0.976725i \(-0.431189\pi\)
0.214496 + 0.976725i \(0.431189\pi\)
\(54\) 2.24621 0.305671
\(55\) −2.56155 −0.345400
\(56\) 0 0
\(57\) −2.87689 −0.381054
\(58\) −8.87689 −1.16559
\(59\) 4.00000 0.520756 0.260378 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) −1.12311 −0.144992
\(61\) 9.36932 1.19962 0.599809 0.800143i \(-0.295243\pi\)
0.599809 + 0.800143i \(0.295243\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 5.56155 0.695194
\(65\) 4.56155 0.565791
\(66\) 10.2462 1.26122
\(67\) −6.24621 −0.763096 −0.381548 0.924349i \(-0.624609\pi\)
−0.381548 + 0.924349i \(0.624609\pi\)
\(68\) 2.00000 0.242536
\(69\) −13.1231 −1.57984
\(70\) 0 0
\(71\) 8.00000 0.949425 0.474713 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) −8.68466 −1.02350
\(73\) −4.24621 −0.496981 −0.248491 0.968634i \(-0.579935\pi\)
−0.248491 + 0.968634i \(0.579935\pi\)
\(74\) 9.36932 1.08916
\(75\) 2.56155 0.295783
\(76\) −0.492423 −0.0564847
\(77\) 0 0
\(78\) −18.2462 −2.06598
\(79\) −6.56155 −0.738232 −0.369116 0.929383i \(-0.620340\pi\)
−0.369116 + 0.929383i \(0.620340\pi\)
\(80\) 4.68466 0.523761
\(81\) −7.00000 −0.777778
\(82\) 4.87689 0.538563
\(83\) −4.00000 −0.439057 −0.219529 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 0 0
\(85\) −4.56155 −0.494770
\(86\) 14.2462 1.53621
\(87\) −14.5616 −1.56116
\(88\) −6.24621 −0.665848
\(89\) −7.12311 −0.755048 −0.377524 0.926000i \(-0.623224\pi\)
−0.377524 + 0.926000i \(0.623224\pi\)
\(90\) −5.56155 −0.586239
\(91\) 0 0
\(92\) −2.24621 −0.234184
\(93\) 0 0
\(94\) −5.75379 −0.593458
\(95\) 1.12311 0.115228
\(96\) −6.24621 −0.637501
\(97\) 14.8078 1.50350 0.751750 0.659448i \(-0.229210\pi\)
0.751750 + 0.659448i \(0.229210\pi\)
\(98\) 0 0
\(99\) 9.12311 0.916907
\(100\) 0.438447 0.0438447
\(101\) −0.246211 −0.0244989 −0.0122495 0.999925i \(-0.503899\pi\)
−0.0122495 + 0.999925i \(0.503899\pi\)
\(102\) 18.2462 1.80664
\(103\) −1.43845 −0.141734 −0.0708672 0.997486i \(-0.522577\pi\)
−0.0708672 + 0.997486i \(0.522577\pi\)
\(104\) 11.1231 1.09071
\(105\) 0 0
\(106\) 4.87689 0.473686
\(107\) −11.3693 −1.09911 −0.549557 0.835456i \(-0.685203\pi\)
−0.549557 + 0.835456i \(0.685203\pi\)
\(108\) 0.630683 0.0606875
\(109\) 17.6847 1.69388 0.846942 0.531686i \(-0.178441\pi\)
0.846942 + 0.531686i \(0.178441\pi\)
\(110\) −4.00000 −0.381385
\(111\) 15.3693 1.45879
\(112\) 0 0
\(113\) −14.0000 −1.31701 −0.658505 0.752577i \(-0.728811\pi\)
−0.658505 + 0.752577i \(0.728811\pi\)
\(114\) −4.49242 −0.420754
\(115\) 5.12311 0.477732
\(116\) −2.49242 −0.231416
\(117\) −16.2462 −1.50196
\(118\) 6.24621 0.575010
\(119\) 0 0
\(120\) 6.24621 0.570198
\(121\) −4.43845 −0.403495
\(122\) 14.6307 1.32460
\(123\) 8.00000 0.721336
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 10.2462 0.909204 0.454602 0.890695i \(-0.349781\pi\)
0.454602 + 0.890695i \(0.349781\pi\)
\(128\) 13.5616 1.19868
\(129\) 23.3693 2.05755
\(130\) 7.12311 0.624738
\(131\) 9.12311 0.797089 0.398545 0.917149i \(-0.369515\pi\)
0.398545 + 0.917149i \(0.369515\pi\)
\(132\) 2.87689 0.250402
\(133\) 0 0
\(134\) −9.75379 −0.842599
\(135\) −1.43845 −0.123802
\(136\) −11.1231 −0.953798
\(137\) −8.87689 −0.758404 −0.379202 0.925314i \(-0.623801\pi\)
−0.379202 + 0.925314i \(0.623801\pi\)
\(138\) −20.4924 −1.74443
\(139\) 6.87689 0.583291 0.291645 0.956527i \(-0.405797\pi\)
0.291645 + 0.956527i \(0.405797\pi\)
\(140\) 0 0
\(141\) −9.43845 −0.794861
\(142\) 12.4924 1.04834
\(143\) −11.6847 −0.977120
\(144\) −16.6847 −1.39039
\(145\) 5.68466 0.472085
\(146\) −6.63068 −0.548759
\(147\) 0 0
\(148\) 2.63068 0.216241
\(149\) −4.24621 −0.347863 −0.173932 0.984758i \(-0.555647\pi\)
−0.173932 + 0.984758i \(0.555647\pi\)
\(150\) 4.00000 0.326599
\(151\) 21.9309 1.78471 0.892354 0.451335i \(-0.149052\pi\)
0.892354 + 0.451335i \(0.149052\pi\)
\(152\) 2.73863 0.222133
\(153\) 16.2462 1.31343
\(154\) 0 0
\(155\) 0 0
\(156\) −5.12311 −0.410177
\(157\) −3.75379 −0.299585 −0.149792 0.988717i \(-0.547861\pi\)
−0.149792 + 0.988717i \(0.547861\pi\)
\(158\) −10.2462 −0.815145
\(159\) 8.00000 0.634441
\(160\) 2.43845 0.192776
\(161\) 0 0
\(162\) −10.9309 −0.858810
\(163\) 1.12311 0.0879684 0.0439842 0.999032i \(-0.485995\pi\)
0.0439842 + 0.999032i \(0.485995\pi\)
\(164\) 1.36932 0.106926
\(165\) −6.56155 −0.510816
\(166\) −6.24621 −0.484800
\(167\) −21.9309 −1.69706 −0.848531 0.529146i \(-0.822512\pi\)
−0.848531 + 0.529146i \(0.822512\pi\)
\(168\) 0 0
\(169\) 7.80776 0.600597
\(170\) −7.12311 −0.546317
\(171\) −4.00000 −0.305888
\(172\) 4.00000 0.304997
\(173\) 8.56155 0.650923 0.325461 0.945555i \(-0.394480\pi\)
0.325461 + 0.945555i \(0.394480\pi\)
\(174\) −22.7386 −1.72381
\(175\) 0 0
\(176\) −12.0000 −0.904534
\(177\) 10.2462 0.770152
\(178\) −11.1231 −0.833712
\(179\) 20.0000 1.49487 0.747435 0.664335i \(-0.231285\pi\)
0.747435 + 0.664335i \(0.231285\pi\)
\(180\) −1.56155 −0.116391
\(181\) −23.6155 −1.75533 −0.877664 0.479276i \(-0.840899\pi\)
−0.877664 + 0.479276i \(0.840899\pi\)
\(182\) 0 0
\(183\) 24.0000 1.77413
\(184\) 12.4924 0.920954
\(185\) −6.00000 −0.441129
\(186\) 0 0
\(187\) 11.6847 0.854467
\(188\) −1.61553 −0.117824
\(189\) 0 0
\(190\) 1.75379 0.127233
\(191\) −9.43845 −0.682942 −0.341471 0.939892i \(-0.610925\pi\)
−0.341471 + 0.939892i \(0.610925\pi\)
\(192\) 14.2462 1.02813
\(193\) −5.36932 −0.386492 −0.193246 0.981150i \(-0.561902\pi\)
−0.193246 + 0.981150i \(0.561902\pi\)
\(194\) 23.1231 1.66014
\(195\) 11.6847 0.836756
\(196\) 0 0
\(197\) −7.12311 −0.507500 −0.253750 0.967270i \(-0.581664\pi\)
−0.253750 + 0.967270i \(0.581664\pi\)
\(198\) 14.2462 1.01243
\(199\) 18.2462 1.29344 0.646720 0.762728i \(-0.276140\pi\)
0.646720 + 0.762728i \(0.276140\pi\)
\(200\) −2.43845 −0.172424
\(201\) −16.0000 −1.12855
\(202\) −0.384472 −0.0270513
\(203\) 0 0
\(204\) 5.12311 0.358689
\(205\) −3.12311 −0.218127
\(206\) −2.24621 −0.156501
\(207\) −18.2462 −1.26820
\(208\) 21.3693 1.48170
\(209\) −2.87689 −0.198999
\(210\) 0 0
\(211\) −23.0540 −1.58710 −0.793551 0.608504i \(-0.791770\pi\)
−0.793551 + 0.608504i \(0.791770\pi\)
\(212\) 1.36932 0.0940451
\(213\) 20.4924 1.40412
\(214\) −17.7538 −1.21362
\(215\) −9.12311 −0.622191
\(216\) −3.50758 −0.238660
\(217\) 0 0
\(218\) 27.6155 1.87036
\(219\) −10.8769 −0.734992
\(220\) −1.12311 −0.0757198
\(221\) −20.8078 −1.39968
\(222\) 24.0000 1.61077
\(223\) 6.56155 0.439394 0.219697 0.975568i \(-0.429493\pi\)
0.219697 + 0.975568i \(0.429493\pi\)
\(224\) 0 0
\(225\) 3.56155 0.237437
\(226\) −21.8617 −1.45422
\(227\) −23.6847 −1.57201 −0.786003 0.618223i \(-0.787853\pi\)
−0.786003 + 0.618223i \(0.787853\pi\)
\(228\) −1.26137 −0.0835360
\(229\) −19.1231 −1.26369 −0.631845 0.775095i \(-0.717702\pi\)
−0.631845 + 0.775095i \(0.717702\pi\)
\(230\) 8.00000 0.527504
\(231\) 0 0
\(232\) 13.8617 0.910068
\(233\) −3.12311 −0.204601 −0.102301 0.994754i \(-0.532620\pi\)
−0.102301 + 0.994754i \(0.532620\pi\)
\(234\) −25.3693 −1.65844
\(235\) 3.68466 0.240361
\(236\) 1.75379 0.114162
\(237\) −16.8078 −1.09178
\(238\) 0 0
\(239\) −0.807764 −0.0522499 −0.0261250 0.999659i \(-0.508317\pi\)
−0.0261250 + 0.999659i \(0.508317\pi\)
\(240\) 12.0000 0.774597
\(241\) −12.2462 −0.788848 −0.394424 0.918929i \(-0.629056\pi\)
−0.394424 + 0.918929i \(0.629056\pi\)
\(242\) −6.93087 −0.445533
\(243\) −22.2462 −1.42710
\(244\) 4.10795 0.262985
\(245\) 0 0
\(246\) 12.4924 0.796488
\(247\) 5.12311 0.325975
\(248\) 0 0
\(249\) −10.2462 −0.649327
\(250\) −1.56155 −0.0987613
\(251\) 17.1231 1.08080 0.540400 0.841408i \(-0.318273\pi\)
0.540400 + 0.841408i \(0.318273\pi\)
\(252\) 0 0
\(253\) −13.1231 −0.825043
\(254\) 16.0000 1.00393
\(255\) −11.6847 −0.731722
\(256\) 10.0540 0.628373
\(257\) −22.4924 −1.40304 −0.701519 0.712650i \(-0.747495\pi\)
−0.701519 + 0.712650i \(0.747495\pi\)
\(258\) 36.4924 2.27192
\(259\) 0 0
\(260\) 2.00000 0.124035
\(261\) −20.2462 −1.25321
\(262\) 14.2462 0.880134
\(263\) −21.1231 −1.30251 −0.651253 0.758860i \(-0.725756\pi\)
−0.651253 + 0.758860i \(0.725756\pi\)
\(264\) −16.0000 −0.984732
\(265\) −3.12311 −0.191851
\(266\) 0 0
\(267\) −18.2462 −1.11665
\(268\) −2.73863 −0.167289
\(269\) −28.7386 −1.75223 −0.876113 0.482106i \(-0.839872\pi\)
−0.876113 + 0.482106i \(0.839872\pi\)
\(270\) −2.24621 −0.136700
\(271\) 16.0000 0.971931 0.485965 0.873978i \(-0.338468\pi\)
0.485965 + 0.873978i \(0.338468\pi\)
\(272\) −21.3693 −1.29571
\(273\) 0 0
\(274\) −13.8617 −0.837418
\(275\) 2.56155 0.154467
\(276\) −5.75379 −0.346337
\(277\) 16.2462 0.976140 0.488070 0.872804i \(-0.337701\pi\)
0.488070 + 0.872804i \(0.337701\pi\)
\(278\) 10.7386 0.644060
\(279\) 0 0
\(280\) 0 0
\(281\) 16.5616 0.987979 0.493990 0.869468i \(-0.335538\pi\)
0.493990 + 0.869468i \(0.335538\pi\)
\(282\) −14.7386 −0.877673
\(283\) 23.6847 1.40791 0.703953 0.710246i \(-0.251416\pi\)
0.703953 + 0.710246i \(0.251416\pi\)
\(284\) 3.50758 0.208136
\(285\) 2.87689 0.170413
\(286\) −18.2462 −1.07892
\(287\) 0 0
\(288\) −8.68466 −0.511748
\(289\) 3.80776 0.223986
\(290\) 8.87689 0.521269
\(291\) 37.9309 2.22355
\(292\) −1.86174 −0.108950
\(293\) −9.68466 −0.565784 −0.282892 0.959152i \(-0.591294\pi\)
−0.282892 + 0.959152i \(0.591294\pi\)
\(294\) 0 0
\(295\) −4.00000 −0.232889
\(296\) −14.6307 −0.850391
\(297\) 3.68466 0.213806
\(298\) −6.63068 −0.384105
\(299\) 23.3693 1.35148
\(300\) 1.12311 0.0648425
\(301\) 0 0
\(302\) 34.2462 1.97065
\(303\) −0.630683 −0.0362318
\(304\) 5.26137 0.301760
\(305\) −9.36932 −0.536486
\(306\) 25.3693 1.45027
\(307\) 31.6847 1.80834 0.904169 0.427174i \(-0.140491\pi\)
0.904169 + 0.427174i \(0.140491\pi\)
\(308\) 0 0
\(309\) −3.68466 −0.209613
\(310\) 0 0
\(311\) 9.61553 0.545247 0.272623 0.962121i \(-0.412109\pi\)
0.272623 + 0.962121i \(0.412109\pi\)
\(312\) 28.4924 1.61307
\(313\) −31.3002 −1.76919 −0.884596 0.466359i \(-0.845566\pi\)
−0.884596 + 0.466359i \(0.845566\pi\)
\(314\) −5.86174 −0.330797
\(315\) 0 0
\(316\) −2.87689 −0.161838
\(317\) −22.4924 −1.26330 −0.631650 0.775254i \(-0.717622\pi\)
−0.631650 + 0.775254i \(0.717622\pi\)
\(318\) 12.4924 0.700540
\(319\) −14.5616 −0.815290
\(320\) −5.56155 −0.310900
\(321\) −29.1231 −1.62549
\(322\) 0 0
\(323\) −5.12311 −0.285057
\(324\) −3.06913 −0.170507
\(325\) −4.56155 −0.253029
\(326\) 1.75379 0.0971334
\(327\) 45.3002 2.50511
\(328\) −7.61553 −0.420497
\(329\) 0 0
\(330\) −10.2462 −0.564035
\(331\) 12.0000 0.659580 0.329790 0.944054i \(-0.393022\pi\)
0.329790 + 0.944054i \(0.393022\pi\)
\(332\) −1.75379 −0.0962517
\(333\) 21.3693 1.17103
\(334\) −34.2462 −1.87387
\(335\) 6.24621 0.341267
\(336\) 0 0
\(337\) −34.4924 −1.87892 −0.939461 0.342656i \(-0.888674\pi\)
−0.939461 + 0.342656i \(0.888674\pi\)
\(338\) 12.1922 0.663170
\(339\) −35.8617 −1.94774
\(340\) −2.00000 −0.108465
\(341\) 0 0
\(342\) −6.24621 −0.337756
\(343\) 0 0
\(344\) −22.2462 −1.19944
\(345\) 13.1231 0.706524
\(346\) 13.3693 0.718739
\(347\) −1.12311 −0.0602915 −0.0301457 0.999546i \(-0.509597\pi\)
−0.0301457 + 0.999546i \(0.509597\pi\)
\(348\) −6.38447 −0.342244
\(349\) 22.4924 1.20399 0.601996 0.798499i \(-0.294372\pi\)
0.601996 + 0.798499i \(0.294372\pi\)
\(350\) 0 0
\(351\) −6.56155 −0.350230
\(352\) −6.24621 −0.332924
\(353\) 14.8078 0.788138 0.394069 0.919081i \(-0.371067\pi\)
0.394069 + 0.919081i \(0.371067\pi\)
\(354\) 16.0000 0.850390
\(355\) −8.00000 −0.424596
\(356\) −3.12311 −0.165524
\(357\) 0 0
\(358\) 31.2311 1.65061
\(359\) 8.00000 0.422224 0.211112 0.977462i \(-0.432292\pi\)
0.211112 + 0.977462i \(0.432292\pi\)
\(360\) 8.68466 0.457722
\(361\) −17.7386 −0.933612
\(362\) −36.8769 −1.93821
\(363\) −11.3693 −0.596734
\(364\) 0 0
\(365\) 4.24621 0.222257
\(366\) 37.4773 1.95897
\(367\) −3.68466 −0.192338 −0.0961688 0.995365i \(-0.530659\pi\)
−0.0961688 + 0.995365i \(0.530659\pi\)
\(368\) 24.0000 1.25109
\(369\) 11.1231 0.579046
\(370\) −9.36932 −0.487088
\(371\) 0 0
\(372\) 0 0
\(373\) 29.3693 1.52069 0.760343 0.649522i \(-0.225031\pi\)
0.760343 + 0.649522i \(0.225031\pi\)
\(374\) 18.2462 0.943489
\(375\) −2.56155 −0.132278
\(376\) 8.98485 0.463358
\(377\) 25.9309 1.33551
\(378\) 0 0
\(379\) 16.4924 0.847159 0.423579 0.905859i \(-0.360773\pi\)
0.423579 + 0.905859i \(0.360773\pi\)
\(380\) 0.492423 0.0252607
\(381\) 26.2462 1.34463
\(382\) −14.7386 −0.754094
\(383\) 10.2462 0.523557 0.261778 0.965128i \(-0.415691\pi\)
0.261778 + 0.965128i \(0.415691\pi\)
\(384\) 34.7386 1.77275
\(385\) 0 0
\(386\) −8.38447 −0.426758
\(387\) 32.4924 1.65168
\(388\) 6.49242 0.329603
\(389\) 3.93087 0.199303 0.0996515 0.995022i \(-0.468227\pi\)
0.0996515 + 0.995022i \(0.468227\pi\)
\(390\) 18.2462 0.923933
\(391\) −23.3693 −1.18184
\(392\) 0 0
\(393\) 23.3693 1.17883
\(394\) −11.1231 −0.560374
\(395\) 6.56155 0.330148
\(396\) 4.00000 0.201008
\(397\) −23.4384 −1.17634 −0.588171 0.808737i \(-0.700152\pi\)
−0.588171 + 0.808737i \(0.700152\pi\)
\(398\) 28.4924 1.42820
\(399\) 0 0
\(400\) −4.68466 −0.234233
\(401\) 27.4384 1.37021 0.685105 0.728444i \(-0.259756\pi\)
0.685105 + 0.728444i \(0.259756\pi\)
\(402\) −24.9848 −1.24613
\(403\) 0 0
\(404\) −0.107951 −0.00537074
\(405\) 7.00000 0.347833
\(406\) 0 0
\(407\) 15.3693 0.761829
\(408\) −28.4924 −1.41059
\(409\) 26.4924 1.30997 0.654983 0.755644i \(-0.272676\pi\)
0.654983 + 0.755644i \(0.272676\pi\)
\(410\) −4.87689 −0.240853
\(411\) −22.7386 −1.12161
\(412\) −0.630683 −0.0310715
\(413\) 0 0
\(414\) −28.4924 −1.40033
\(415\) 4.00000 0.196352
\(416\) 11.1231 0.545355
\(417\) 17.6155 0.862636
\(418\) −4.49242 −0.219732
\(419\) −9.75379 −0.476504 −0.238252 0.971203i \(-0.576574\pi\)
−0.238252 + 0.971203i \(0.576574\pi\)
\(420\) 0 0
\(421\) 9.68466 0.472001 0.236001 0.971753i \(-0.424163\pi\)
0.236001 + 0.971753i \(0.424163\pi\)
\(422\) −36.0000 −1.75245
\(423\) −13.1231 −0.638067
\(424\) −7.61553 −0.369843
\(425\) 4.56155 0.221268
\(426\) 32.0000 1.55041
\(427\) 0 0
\(428\) −4.98485 −0.240952
\(429\) −29.9309 −1.44508
\(430\) −14.2462 −0.687013
\(431\) 0.807764 0.0389086 0.0194543 0.999811i \(-0.493807\pi\)
0.0194543 + 0.999811i \(0.493807\pi\)
\(432\) −6.73863 −0.324213
\(433\) 8.24621 0.396288 0.198144 0.980173i \(-0.436509\pi\)
0.198144 + 0.980173i \(0.436509\pi\)
\(434\) 0 0
\(435\) 14.5616 0.698173
\(436\) 7.75379 0.371339
\(437\) 5.75379 0.275241
\(438\) −16.9848 −0.811567
\(439\) 15.3693 0.733537 0.366769 0.930312i \(-0.380464\pi\)
0.366769 + 0.930312i \(0.380464\pi\)
\(440\) 6.24621 0.297776
\(441\) 0 0
\(442\) −32.4924 −1.54551
\(443\) −27.3693 −1.30036 −0.650178 0.759782i \(-0.725306\pi\)
−0.650178 + 0.759782i \(0.725306\pi\)
\(444\) 6.73863 0.319801
\(445\) 7.12311 0.337668
\(446\) 10.2462 0.485172
\(447\) −10.8769 −0.514459
\(448\) 0 0
\(449\) 18.8078 0.887593 0.443797 0.896128i \(-0.353631\pi\)
0.443797 + 0.896128i \(0.353631\pi\)
\(450\) 5.56155 0.262174
\(451\) 8.00000 0.376705
\(452\) −6.13826 −0.288719
\(453\) 56.1771 2.63943
\(454\) −36.9848 −1.73578
\(455\) 0 0
\(456\) 7.01515 0.328515
\(457\) −8.87689 −0.415244 −0.207622 0.978209i \(-0.566572\pi\)
−0.207622 + 0.978209i \(0.566572\pi\)
\(458\) −29.8617 −1.39535
\(459\) 6.56155 0.306267
\(460\) 2.24621 0.104730
\(461\) 4.87689 0.227140 0.113570 0.993530i \(-0.463771\pi\)
0.113570 + 0.993530i \(0.463771\pi\)
\(462\) 0 0
\(463\) −20.4924 −0.952364 −0.476182 0.879347i \(-0.657980\pi\)
−0.476182 + 0.879347i \(0.657980\pi\)
\(464\) 26.6307 1.23630
\(465\) 0 0
\(466\) −4.87689 −0.225918
\(467\) −26.5616 −1.22912 −0.614561 0.788869i \(-0.710667\pi\)
−0.614561 + 0.788869i \(0.710667\pi\)
\(468\) −7.12311 −0.329266
\(469\) 0 0
\(470\) 5.75379 0.265402
\(471\) −9.61553 −0.443060
\(472\) −9.75379 −0.448955
\(473\) 23.3693 1.07452
\(474\) −26.2462 −1.20553
\(475\) −1.12311 −0.0515316
\(476\) 0 0
\(477\) 11.1231 0.509292
\(478\) −1.26137 −0.0576935
\(479\) −13.1231 −0.599610 −0.299805 0.954001i \(-0.596922\pi\)
−0.299805 + 0.954001i \(0.596922\pi\)
\(480\) 6.24621 0.285099
\(481\) −27.3693 −1.24793
\(482\) −19.1231 −0.871034
\(483\) 0 0
\(484\) −1.94602 −0.0884557
\(485\) −14.8078 −0.672386
\(486\) −34.7386 −1.57578
\(487\) 5.12311 0.232150 0.116075 0.993240i \(-0.462969\pi\)
0.116075 + 0.993240i \(0.462969\pi\)
\(488\) −22.8466 −1.03422
\(489\) 2.87689 0.130098
\(490\) 0 0
\(491\) 4.17708 0.188509 0.0942545 0.995548i \(-0.469953\pi\)
0.0942545 + 0.995548i \(0.469953\pi\)
\(492\) 3.50758 0.158134
\(493\) −25.9309 −1.16787
\(494\) 8.00000 0.359937
\(495\) −9.12311 −0.410053
\(496\) 0 0
\(497\) 0 0
\(498\) −16.0000 −0.716977
\(499\) −4.17708 −0.186992 −0.0934959 0.995620i \(-0.529804\pi\)
−0.0934959 + 0.995620i \(0.529804\pi\)
\(500\) −0.438447 −0.0196080
\(501\) −56.1771 −2.50981
\(502\) 26.7386 1.19340
\(503\) −10.0691 −0.448960 −0.224480 0.974479i \(-0.572068\pi\)
−0.224480 + 0.974479i \(0.572068\pi\)
\(504\) 0 0
\(505\) 0.246211 0.0109563
\(506\) −20.4924 −0.910999
\(507\) 20.0000 0.888231
\(508\) 4.49242 0.199319
\(509\) 28.2462 1.25199 0.625996 0.779827i \(-0.284693\pi\)
0.625996 + 0.779827i \(0.284693\pi\)
\(510\) −18.2462 −0.807956
\(511\) 0 0
\(512\) −11.4233 −0.504843
\(513\) −1.61553 −0.0713273
\(514\) −35.1231 −1.54921
\(515\) 1.43845 0.0633856
\(516\) 10.2462 0.451064
\(517\) −9.43845 −0.415102
\(518\) 0 0
\(519\) 21.9309 0.962658
\(520\) −11.1231 −0.487780
\(521\) −10.0000 −0.438108 −0.219054 0.975713i \(-0.570297\pi\)
−0.219054 + 0.975713i \(0.570297\pi\)
\(522\) −31.6155 −1.38377
\(523\) −7.50758 −0.328283 −0.164142 0.986437i \(-0.552485\pi\)
−0.164142 + 0.986437i \(0.552485\pi\)
\(524\) 4.00000 0.174741
\(525\) 0 0
\(526\) −32.9848 −1.43821
\(527\) 0 0
\(528\) −30.7386 −1.33773
\(529\) 3.24621 0.141140
\(530\) −4.87689 −0.211839
\(531\) 14.2462 0.618233
\(532\) 0 0
\(533\) −14.2462 −0.617072
\(534\) −28.4924 −1.23299
\(535\) 11.3693 0.491538
\(536\) 15.2311 0.657881
\(537\) 51.2311 2.21078
\(538\) −44.8769 −1.93478
\(539\) 0 0
\(540\) −0.630683 −0.0271403
\(541\) −17.1922 −0.739152 −0.369576 0.929201i \(-0.620497\pi\)
−0.369576 + 0.929201i \(0.620497\pi\)
\(542\) 24.9848 1.07319
\(543\) −60.4924 −2.59598
\(544\) −11.1231 −0.476899
\(545\) −17.6847 −0.757528
\(546\) 0 0
\(547\) 14.2462 0.609124 0.304562 0.952493i \(-0.401490\pi\)
0.304562 + 0.952493i \(0.401490\pi\)
\(548\) −3.89205 −0.166260
\(549\) 33.3693 1.42417
\(550\) 4.00000 0.170561
\(551\) 6.38447 0.271988
\(552\) 32.0000 1.36201
\(553\) 0 0
\(554\) 25.3693 1.07784
\(555\) −15.3693 −0.652391
\(556\) 3.01515 0.127871
\(557\) −4.87689 −0.206641 −0.103320 0.994648i \(-0.532947\pi\)
−0.103320 + 0.994648i \(0.532947\pi\)
\(558\) 0 0
\(559\) −41.6155 −1.76015
\(560\) 0 0
\(561\) 29.9309 1.26368
\(562\) 25.8617 1.09091
\(563\) 28.0000 1.18006 0.590030 0.807382i \(-0.299116\pi\)
0.590030 + 0.807382i \(0.299116\pi\)
\(564\) −4.13826 −0.174252
\(565\) 14.0000 0.588984
\(566\) 36.9848 1.55459
\(567\) 0 0
\(568\) −19.5076 −0.818520
\(569\) 34.9848 1.46664 0.733320 0.679883i \(-0.237969\pi\)
0.733320 + 0.679883i \(0.237969\pi\)
\(570\) 4.49242 0.188167
\(571\) 7.50758 0.314182 0.157091 0.987584i \(-0.449788\pi\)
0.157091 + 0.987584i \(0.449788\pi\)
\(572\) −5.12311 −0.214208
\(573\) −24.1771 −1.01001
\(574\) 0 0
\(575\) −5.12311 −0.213648
\(576\) 19.8078 0.825324
\(577\) −13.0540 −0.543444 −0.271722 0.962376i \(-0.587593\pi\)
−0.271722 + 0.962376i \(0.587593\pi\)
\(578\) 5.94602 0.247322
\(579\) −13.7538 −0.571588
\(580\) 2.49242 0.103492
\(581\) 0 0
\(582\) 59.2311 2.45521
\(583\) 8.00000 0.331326
\(584\) 10.3542 0.428458
\(585\) 16.2462 0.671698
\(586\) −15.1231 −0.624730
\(587\) 9.75379 0.402582 0.201291 0.979531i \(-0.435486\pi\)
0.201291 + 0.979531i \(0.435486\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) −6.24621 −0.257152
\(591\) −18.2462 −0.750549
\(592\) −28.1080 −1.15523
\(593\) 23.4384 0.962502 0.481251 0.876583i \(-0.340183\pi\)
0.481251 + 0.876583i \(0.340183\pi\)
\(594\) 5.75379 0.236081
\(595\) 0 0
\(596\) −1.86174 −0.0762598
\(597\) 46.7386 1.91288
\(598\) 36.4924 1.49229
\(599\) 8.80776 0.359875 0.179938 0.983678i \(-0.442410\pi\)
0.179938 + 0.983678i \(0.442410\pi\)
\(600\) −6.24621 −0.255001
\(601\) 26.4924 1.08065 0.540324 0.841457i \(-0.318302\pi\)
0.540324 + 0.841457i \(0.318302\pi\)
\(602\) 0 0
\(603\) −22.2462 −0.905936
\(604\) 9.61553 0.391250
\(605\) 4.43845 0.180449
\(606\) −0.984845 −0.0400066
\(607\) 4.94602 0.200753 0.100376 0.994950i \(-0.467995\pi\)
0.100376 + 0.994950i \(0.467995\pi\)
\(608\) 2.73863 0.111066
\(609\) 0 0
\(610\) −14.6307 −0.592379
\(611\) 16.8078 0.679969
\(612\) 7.12311 0.287934
\(613\) −8.73863 −0.352950 −0.176475 0.984305i \(-0.556469\pi\)
−0.176475 + 0.984305i \(0.556469\pi\)
\(614\) 49.4773 1.99674
\(615\) −8.00000 −0.322591
\(616\) 0 0
\(617\) 15.7538 0.634224 0.317112 0.948388i \(-0.397287\pi\)
0.317112 + 0.948388i \(0.397287\pi\)
\(618\) −5.75379 −0.231451
\(619\) 42.1080 1.69246 0.846231 0.532817i \(-0.178866\pi\)
0.846231 + 0.532817i \(0.178866\pi\)
\(620\) 0 0
\(621\) −7.36932 −0.295720
\(622\) 15.0152 0.602053
\(623\) 0 0
\(624\) 54.7386 2.19130
\(625\) 1.00000 0.0400000
\(626\) −48.8769 −1.95351
\(627\) −7.36932 −0.294302
\(628\) −1.64584 −0.0656761
\(629\) 27.3693 1.09129
\(630\) 0 0
\(631\) 8.80776 0.350632 0.175316 0.984512i \(-0.443905\pi\)
0.175316 + 0.984512i \(0.443905\pi\)
\(632\) 16.0000 0.636446
\(633\) −59.0540 −2.34718
\(634\) −35.1231 −1.39492
\(635\) −10.2462 −0.406608
\(636\) 3.50758 0.139084
\(637\) 0 0
\(638\) −22.7386 −0.900231
\(639\) 28.4924 1.12714
\(640\) −13.5616 −0.536067
\(641\) 2.00000 0.0789953 0.0394976 0.999220i \(-0.487424\pi\)
0.0394976 + 0.999220i \(0.487424\pi\)
\(642\) −45.4773 −1.79484
\(643\) 2.56155 0.101018 0.0505089 0.998724i \(-0.483916\pi\)
0.0505089 + 0.998724i \(0.483916\pi\)
\(644\) 0 0
\(645\) −23.3693 −0.920166
\(646\) −8.00000 −0.314756
\(647\) −3.50758 −0.137897 −0.0689486 0.997620i \(-0.521964\pi\)
−0.0689486 + 0.997620i \(0.521964\pi\)
\(648\) 17.0691 0.670539
\(649\) 10.2462 0.402199
\(650\) −7.12311 −0.279391
\(651\) 0 0
\(652\) 0.492423 0.0192848
\(653\) 49.2311 1.92656 0.963280 0.268499i \(-0.0865275\pi\)
0.963280 + 0.268499i \(0.0865275\pi\)
\(654\) 70.7386 2.76610
\(655\) −9.12311 −0.356469
\(656\) −14.6307 −0.571232
\(657\) −15.1231 −0.590009
\(658\) 0 0
\(659\) −36.1771 −1.40926 −0.704629 0.709575i \(-0.748887\pi\)
−0.704629 + 0.709575i \(0.748887\pi\)
\(660\) −2.87689 −0.111983
\(661\) −3.12311 −0.121475 −0.0607374 0.998154i \(-0.519345\pi\)
−0.0607374 + 0.998154i \(0.519345\pi\)
\(662\) 18.7386 0.728298
\(663\) −53.3002 −2.07001
\(664\) 9.75379 0.378520
\(665\) 0 0
\(666\) 33.3693 1.29303
\(667\) 29.1231 1.12765
\(668\) −9.61553 −0.372036
\(669\) 16.8078 0.649826
\(670\) 9.75379 0.376822
\(671\) 24.0000 0.926510
\(672\) 0 0
\(673\) −25.8617 −0.996897 −0.498448 0.866919i \(-0.666097\pi\)
−0.498448 + 0.866919i \(0.666097\pi\)
\(674\) −53.8617 −2.07468
\(675\) 1.43845 0.0553659
\(676\) 3.42329 0.131665
\(677\) 23.9309 0.919738 0.459869 0.887987i \(-0.347896\pi\)
0.459869 + 0.887987i \(0.347896\pi\)
\(678\) −56.0000 −2.15067
\(679\) 0 0
\(680\) 11.1231 0.426552
\(681\) −60.6695 −2.32486
\(682\) 0 0
\(683\) 42.7386 1.63535 0.817674 0.575681i \(-0.195263\pi\)
0.817674 + 0.575681i \(0.195263\pi\)
\(684\) −1.75379 −0.0670578
\(685\) 8.87689 0.339169
\(686\) 0 0
\(687\) −48.9848 −1.86889
\(688\) −42.7386 −1.62940
\(689\) −14.2462 −0.542737
\(690\) 20.4924 0.780133
\(691\) −8.49242 −0.323067 −0.161533 0.986867i \(-0.551644\pi\)
−0.161533 + 0.986867i \(0.551644\pi\)
\(692\) 3.75379 0.142698
\(693\) 0 0
\(694\) −1.75379 −0.0665729
\(695\) −6.87689 −0.260855
\(696\) 35.5076 1.34591
\(697\) 14.2462 0.539614
\(698\) 35.1231 1.32943
\(699\) −8.00000 −0.302588
\(700\) 0 0
\(701\) 0.0691303 0.00261102 0.00130551 0.999999i \(-0.499584\pi\)
0.00130551 + 0.999999i \(0.499584\pi\)
\(702\) −10.2462 −0.386718
\(703\) −6.73863 −0.254152
\(704\) 14.2462 0.536924
\(705\) 9.43845 0.355472
\(706\) 23.1231 0.870250
\(707\) 0 0
\(708\) 4.49242 0.168836
\(709\) −18.1771 −0.682655 −0.341327 0.939945i \(-0.610876\pi\)
−0.341327 + 0.939945i \(0.610876\pi\)
\(710\) −12.4924 −0.468832
\(711\) −23.3693 −0.876418
\(712\) 17.3693 0.650943
\(713\) 0 0
\(714\) 0 0
\(715\) 11.6847 0.436981
\(716\) 8.76894 0.327711
\(717\) −2.06913 −0.0772731
\(718\) 12.4924 0.466213
\(719\) −49.6155 −1.85035 −0.925173 0.379544i \(-0.876081\pi\)
−0.925173 + 0.379544i \(0.876081\pi\)
\(720\) 16.6847 0.621801
\(721\) 0 0
\(722\) −27.6998 −1.03088
\(723\) −31.3693 −1.16664
\(724\) −10.3542 −0.384809
\(725\) −5.68466 −0.211123
\(726\) −17.7538 −0.658905
\(727\) −19.5076 −0.723496 −0.361748 0.932276i \(-0.617820\pi\)
−0.361748 + 0.932276i \(0.617820\pi\)
\(728\) 0 0
\(729\) −35.9848 −1.33277
\(730\) 6.63068 0.245413
\(731\) 41.6155 1.53921
\(732\) 10.5227 0.388931
\(733\) 5.68466 0.209968 0.104984 0.994474i \(-0.466521\pi\)
0.104984 + 0.994474i \(0.466521\pi\)
\(734\) −5.75379 −0.212376
\(735\) 0 0
\(736\) 12.4924 0.460477
\(737\) −16.0000 −0.589368
\(738\) 17.3693 0.639373
\(739\) 6.06913 0.223257 0.111628 0.993750i \(-0.464393\pi\)
0.111628 + 0.993750i \(0.464393\pi\)
\(740\) −2.63068 −0.0967058
\(741\) 13.1231 0.482089
\(742\) 0 0
\(743\) 32.9848 1.21010 0.605048 0.796189i \(-0.293154\pi\)
0.605048 + 0.796189i \(0.293154\pi\)
\(744\) 0 0
\(745\) 4.24621 0.155569
\(746\) 45.8617 1.67912
\(747\) −14.2462 −0.521242
\(748\) 5.12311 0.187319
\(749\) 0 0
\(750\) −4.00000 −0.146059
\(751\) 45.9309 1.67604 0.838021 0.545639i \(-0.183713\pi\)
0.838021 + 0.545639i \(0.183713\pi\)
\(752\) 17.2614 0.629457
\(753\) 43.8617 1.59841
\(754\) 40.4924 1.47465
\(755\) −21.9309 −0.798146
\(756\) 0 0
\(757\) 14.6307 0.531761 0.265881 0.964006i \(-0.414337\pi\)
0.265881 + 0.964006i \(0.414337\pi\)
\(758\) 25.7538 0.935420
\(759\) −33.6155 −1.22017
\(760\) −2.73863 −0.0993407
\(761\) −31.7538 −1.15107 −0.575537 0.817776i \(-0.695207\pi\)
−0.575537 + 0.817776i \(0.695207\pi\)
\(762\) 40.9848 1.48472
\(763\) 0 0
\(764\) −4.13826 −0.149717
\(765\) −16.2462 −0.587383
\(766\) 16.0000 0.578103
\(767\) −18.2462 −0.658833
\(768\) 25.7538 0.929310
\(769\) 9.50758 0.342852 0.171426 0.985197i \(-0.445163\pi\)
0.171426 + 0.985197i \(0.445163\pi\)
\(770\) 0 0
\(771\) −57.6155 −2.07497
\(772\) −2.35416 −0.0847281
\(773\) −8.06913 −0.290226 −0.145113 0.989415i \(-0.546355\pi\)
−0.145113 + 0.989415i \(0.546355\pi\)
\(774\) 50.7386 1.82376
\(775\) 0 0
\(776\) −36.1080 −1.29620
\(777\) 0 0
\(778\) 6.13826 0.220067
\(779\) −3.50758 −0.125672
\(780\) 5.12311 0.183437
\(781\) 20.4924 0.733277
\(782\) −36.4924 −1.30497
\(783\) −8.17708 −0.292225
\(784\) 0 0
\(785\) 3.75379 0.133978
\(786\) 36.4924 1.30164
\(787\) 3.82292 0.136272 0.0681362 0.997676i \(-0.478295\pi\)
0.0681362 + 0.997676i \(0.478295\pi\)
\(788\) −3.12311 −0.111256
\(789\) −54.1080 −1.92629
\(790\) 10.2462 0.364544
\(791\) 0 0
\(792\) −22.2462 −0.790485
\(793\) −42.7386 −1.51769
\(794\) −36.6004 −1.29890
\(795\) −8.00000 −0.283731
\(796\) 8.00000 0.283552
\(797\) 13.0540 0.462396 0.231198 0.972907i \(-0.425736\pi\)
0.231198 + 0.972907i \(0.425736\pi\)
\(798\) 0 0
\(799\) −16.8078 −0.594616
\(800\) −2.43845 −0.0862121
\(801\) −25.3693 −0.896381
\(802\) 42.8466 1.51297
\(803\) −10.8769 −0.383837
\(804\) −7.01515 −0.247405
\(805\) 0 0
\(806\) 0 0
\(807\) −73.6155 −2.59139
\(808\) 0.600373 0.0211211
\(809\) −53.5464 −1.88259 −0.941296 0.337584i \(-0.890390\pi\)
−0.941296 + 0.337584i \(0.890390\pi\)
\(810\) 10.9309 0.384072
\(811\) 21.6155 0.759024 0.379512 0.925187i \(-0.376092\pi\)
0.379512 + 0.925187i \(0.376092\pi\)
\(812\) 0 0
\(813\) 40.9848 1.43740
\(814\) 24.0000 0.841200
\(815\) −1.12311 −0.0393407
\(816\) −54.7386 −1.91624
\(817\) −10.2462 −0.358470
\(818\) 41.3693 1.44644
\(819\) 0 0
\(820\) −1.36932 −0.0478186
\(821\) 40.4233 1.41078 0.705391 0.708818i \(-0.250771\pi\)
0.705391 + 0.708818i \(0.250771\pi\)
\(822\) −35.5076 −1.23847
\(823\) −3.50758 −0.122266 −0.0611332 0.998130i \(-0.519471\pi\)
−0.0611332 + 0.998130i \(0.519471\pi\)
\(824\) 3.50758 0.122192
\(825\) 6.56155 0.228444
\(826\) 0 0
\(827\) 19.3693 0.673537 0.336769 0.941587i \(-0.390666\pi\)
0.336769 + 0.941587i \(0.390666\pi\)
\(828\) −8.00000 −0.278019
\(829\) −43.1231 −1.49773 −0.748864 0.662724i \(-0.769400\pi\)
−0.748864 + 0.662724i \(0.769400\pi\)
\(830\) 6.24621 0.216809
\(831\) 41.6155 1.44363
\(832\) −25.3693 −0.879523
\(833\) 0 0
\(834\) 27.5076 0.952510
\(835\) 21.9309 0.758949
\(836\) −1.26137 −0.0436253
\(837\) 0 0
\(838\) −15.2311 −0.526148
\(839\) 37.1231 1.28163 0.640816 0.767695i \(-0.278596\pi\)
0.640816 + 0.767695i \(0.278596\pi\)
\(840\) 0 0
\(841\) 3.31534 0.114322
\(842\) 15.1231 0.521177
\(843\) 42.4233 1.46114
\(844\) −10.1080 −0.347930
\(845\) −7.80776 −0.268595
\(846\) −20.4924 −0.704544
\(847\) 0 0
\(848\) −14.6307 −0.502420
\(849\) 60.6695 2.08217
\(850\) 7.12311 0.244321
\(851\) −30.7386 −1.05371
\(852\) 8.98485 0.307816
\(853\) 56.7386 1.94269 0.971347 0.237666i \(-0.0763824\pi\)
0.971347 + 0.237666i \(0.0763824\pi\)
\(854\) 0 0
\(855\) 4.00000 0.136797
\(856\) 27.7235 0.947569
\(857\) 32.2462 1.10151 0.550755 0.834667i \(-0.314340\pi\)
0.550755 + 0.834667i \(0.314340\pi\)
\(858\) −46.7386 −1.59563
\(859\) −16.4924 −0.562714 −0.281357 0.959603i \(-0.590785\pi\)
−0.281357 + 0.959603i \(0.590785\pi\)
\(860\) −4.00000 −0.136399
\(861\) 0 0
\(862\) 1.26137 0.0429623
\(863\) −42.2462 −1.43808 −0.719039 0.694970i \(-0.755418\pi\)
−0.719039 + 0.694970i \(0.755418\pi\)
\(864\) −3.50758 −0.119330
\(865\) −8.56155 −0.291102
\(866\) 12.8769 0.437575
\(867\) 9.75379 0.331256
\(868\) 0 0
\(869\) −16.8078 −0.570164
\(870\) 22.7386 0.770912
\(871\) 28.4924 0.965429
\(872\) −43.1231 −1.46033
\(873\) 52.7386 1.78493
\(874\) 8.98485 0.303917
\(875\) 0 0
\(876\) −4.76894 −0.161128
\(877\) −23.7538 −0.802108 −0.401054 0.916054i \(-0.631356\pi\)
−0.401054 + 0.916054i \(0.631356\pi\)
\(878\) 24.0000 0.809961
\(879\) −24.8078 −0.836745
\(880\) 12.0000 0.404520
\(881\) −45.8617 −1.54512 −0.772561 0.634941i \(-0.781024\pi\)
−0.772561 + 0.634941i \(0.781024\pi\)
\(882\) 0 0
\(883\) 24.4924 0.824236 0.412118 0.911131i \(-0.364789\pi\)
0.412118 + 0.911131i \(0.364789\pi\)
\(884\) −9.12311 −0.306843
\(885\) −10.2462 −0.344423
\(886\) −42.7386 −1.43583
\(887\) 12.4924 0.419454 0.209727 0.977760i \(-0.432742\pi\)
0.209727 + 0.977760i \(0.432742\pi\)
\(888\) −37.4773 −1.25765
\(889\) 0 0
\(890\) 11.1231 0.372847
\(891\) −17.9309 −0.600707
\(892\) 2.87689 0.0963255
\(893\) 4.13826 0.138482
\(894\) −16.9848 −0.568058
\(895\) −20.0000 −0.668526
\(896\) 0 0
\(897\) 59.8617 1.99873
\(898\) 29.3693 0.980067
\(899\) 0 0
\(900\) 1.56155 0.0520518
\(901\) 14.2462 0.474610
\(902\) 12.4924 0.415952
\(903\) 0 0
\(904\) 34.1383 1.13542
\(905\) 23.6155 0.785007
\(906\) 87.7235 2.91442
\(907\) 50.1080 1.66381 0.831904 0.554920i \(-0.187251\pi\)
0.831904 + 0.554920i \(0.187251\pi\)
\(908\) −10.3845 −0.344621
\(909\) −0.876894 −0.0290848
\(910\) 0 0
\(911\) 4.49242 0.148841 0.0744203 0.997227i \(-0.476289\pi\)
0.0744203 + 0.997227i \(0.476289\pi\)
\(912\) 13.4773 0.446277
\(913\) −10.2462 −0.339100
\(914\) −13.8617 −0.458506
\(915\) −24.0000 −0.793416
\(916\) −8.38447 −0.277031
\(917\) 0 0
\(918\) 10.2462 0.338175
\(919\) −13.3002 −0.438733 −0.219366 0.975643i \(-0.570399\pi\)
−0.219366 + 0.975643i \(0.570399\pi\)
\(920\) −12.4924 −0.411863
\(921\) 81.1619 2.67438
\(922\) 7.61553 0.250804
\(923\) −36.4924 −1.20116
\(924\) 0 0
\(925\) 6.00000 0.197279
\(926\) −32.0000 −1.05159
\(927\) −5.12311 −0.168265
\(928\) 13.8617 0.455034
\(929\) 52.1080 1.70961 0.854803 0.518952i \(-0.173678\pi\)
0.854803 + 0.518952i \(0.173678\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −1.36932 −0.0448535
\(933\) 24.6307 0.806372
\(934\) −41.4773 −1.35718
\(935\) −11.6847 −0.382129
\(936\) 39.6155 1.29487
\(937\) −22.6695 −0.740580 −0.370290 0.928916i \(-0.620742\pi\)
−0.370290 + 0.928916i \(0.620742\pi\)
\(938\) 0 0
\(939\) −80.1771 −2.61648
\(940\) 1.61553 0.0526927
\(941\) 13.8617 0.451880 0.225940 0.974141i \(-0.427455\pi\)
0.225940 + 0.974141i \(0.427455\pi\)
\(942\) −15.0152 −0.489220
\(943\) −16.0000 −0.521032
\(944\) −18.7386 −0.609891
\(945\) 0 0
\(946\) 36.4924 1.18647
\(947\) 4.00000 0.129983 0.0649913 0.997886i \(-0.479298\pi\)
0.0649913 + 0.997886i \(0.479298\pi\)
\(948\) −7.36932 −0.239344
\(949\) 19.3693 0.628755
\(950\) −1.75379 −0.0569004
\(951\) −57.6155 −1.86831
\(952\) 0 0
\(953\) −24.8769 −0.805842 −0.402921 0.915235i \(-0.632005\pi\)
−0.402921 + 0.915235i \(0.632005\pi\)
\(954\) 17.3693 0.562352
\(955\) 9.43845 0.305421
\(956\) −0.354162 −0.0114544
\(957\) −37.3002 −1.20574
\(958\) −20.4924 −0.662080
\(959\) 0 0
\(960\) −14.2462 −0.459794
\(961\) −31.0000 −1.00000
\(962\) −42.7386 −1.37795
\(963\) −40.4924 −1.30485
\(964\) −5.36932 −0.172934
\(965\) 5.36932 0.172844
\(966\) 0 0
\(967\) −26.8769 −0.864303 −0.432151 0.901801i \(-0.642245\pi\)
−0.432151 + 0.901801i \(0.642245\pi\)
\(968\) 10.8229 0.347862
\(969\) −13.1231 −0.421575
\(970\) −23.1231 −0.742438
\(971\) 49.4773 1.58780 0.793901 0.608048i \(-0.208047\pi\)
0.793901 + 0.608048i \(0.208047\pi\)
\(972\) −9.75379 −0.312853
\(973\) 0 0
\(974\) 8.00000 0.256337
\(975\) −11.6847 −0.374209
\(976\) −43.8920 −1.40495
\(977\) −49.2311 −1.57504 −0.787521 0.616288i \(-0.788636\pi\)
−0.787521 + 0.616288i \(0.788636\pi\)
\(978\) 4.49242 0.143652
\(979\) −18.2462 −0.583151
\(980\) 0 0
\(981\) 62.9848 2.01095
\(982\) 6.52273 0.208149
\(983\) 10.4233 0.332451 0.166226 0.986088i \(-0.446842\pi\)
0.166226 + 0.986088i \(0.446842\pi\)
\(984\) −19.5076 −0.621879
\(985\) 7.12311 0.226961
\(986\) −40.4924 −1.28954
\(987\) 0 0
\(988\) 2.24621 0.0714615
\(989\) −46.7386 −1.48620
\(990\) −14.2462 −0.452774
\(991\) −20.4924 −0.650963 −0.325482 0.945548i \(-0.605526\pi\)
−0.325482 + 0.945548i \(0.605526\pi\)
\(992\) 0 0
\(993\) 30.7386 0.975461
\(994\) 0 0
\(995\) −18.2462 −0.578444
\(996\) −4.49242 −0.142348
\(997\) −9.68466 −0.306716 −0.153358 0.988171i \(-0.549009\pi\)
−0.153358 + 0.988171i \(0.549009\pi\)
\(998\) −6.52273 −0.206473
\(999\) 8.63068 0.273063
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 245.2.a.d.1.2 2
3.2 odd 2 2205.2.a.x.1.1 2
4.3 odd 2 3920.2.a.bs.1.1 2
5.2 odd 4 1225.2.b.f.99.3 4
5.3 odd 4 1225.2.b.f.99.2 4
5.4 even 2 1225.2.a.s.1.1 2
7.2 even 3 245.2.e.h.116.1 4
7.3 odd 6 245.2.e.i.226.1 4
7.4 even 3 245.2.e.h.226.1 4
7.5 odd 6 245.2.e.i.116.1 4
7.6 odd 2 35.2.a.b.1.2 2
21.20 even 2 315.2.a.e.1.1 2
28.27 even 2 560.2.a.i.1.2 2
35.13 even 4 175.2.b.b.99.2 4
35.27 even 4 175.2.b.b.99.3 4
35.34 odd 2 175.2.a.f.1.1 2
56.13 odd 2 2240.2.a.bh.1.2 2
56.27 even 2 2240.2.a.bd.1.1 2
77.76 even 2 4235.2.a.m.1.1 2
84.83 odd 2 5040.2.a.bt.1.2 2
91.90 odd 2 5915.2.a.l.1.1 2
105.62 odd 4 1575.2.d.e.1324.2 4
105.83 odd 4 1575.2.d.e.1324.3 4
105.104 even 2 1575.2.a.p.1.2 2
140.27 odd 4 2800.2.g.t.449.1 4
140.83 odd 4 2800.2.g.t.449.4 4
140.139 even 2 2800.2.a.bi.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.2.a.b.1.2 2 7.6 odd 2
175.2.a.f.1.1 2 35.34 odd 2
175.2.b.b.99.2 4 35.13 even 4
175.2.b.b.99.3 4 35.27 even 4
245.2.a.d.1.2 2 1.1 even 1 trivial
245.2.e.h.116.1 4 7.2 even 3
245.2.e.h.226.1 4 7.4 even 3
245.2.e.i.116.1 4 7.5 odd 6
245.2.e.i.226.1 4 7.3 odd 6
315.2.a.e.1.1 2 21.20 even 2
560.2.a.i.1.2 2 28.27 even 2
1225.2.a.s.1.1 2 5.4 even 2
1225.2.b.f.99.2 4 5.3 odd 4
1225.2.b.f.99.3 4 5.2 odd 4
1575.2.a.p.1.2 2 105.104 even 2
1575.2.d.e.1324.2 4 105.62 odd 4
1575.2.d.e.1324.3 4 105.83 odd 4
2205.2.a.x.1.1 2 3.2 odd 2
2240.2.a.bd.1.1 2 56.27 even 2
2240.2.a.bh.1.2 2 56.13 odd 2
2800.2.a.bi.1.1 2 140.139 even 2
2800.2.g.t.449.1 4 140.27 odd 4
2800.2.g.t.449.4 4 140.83 odd 4
3920.2.a.bs.1.1 2 4.3 odd 2
4235.2.a.m.1.1 2 77.76 even 2
5040.2.a.bt.1.2 2 84.83 odd 2
5915.2.a.l.1.1 2 91.90 odd 2